Vibration-based damage detection in anisotropic laminated composite beams by a shear deformable nite element and harmony search optimization

UntitledVibration-based damage detection in anisotropic laminated composite beams by a shear deformable nite element and harmony search optimization Volkan Kahya  ( [email protected] )
Karadeniz Technical University Sebahat imek 
Karadeniz Technical University Vedat Toan 
Karadeniz Technical University
Posted Date: May 23rd, 2022
DOI: https://doi.org/10.21203/rs.3.rs-1681275/v1
License: This work is licensed under a Creative Commons Attribution 4.0 International License.   Read Full License
deformable finite element and harmony search optimization
Volkan Kahya
61080 Trabzon, Türkiye, E-mail: [email protected]
Sebahat imek
61080 Trabzon, Türkiye, E-mail: [email protected]
Vedat Toan
61080 Trabzon, Türkiye, E-mail: [email protected]
Corresponding author
vibration characteristics, they can be considered as the diagnostic parameters with the help of an optimization
technique. For the present study, frequencies and mode shapes are used as the damage diagnostic parameters, and
the harmony search algorithm (HSA) is used as the optimization tool. For finite element (FE) analyses, a five-
node, 13-degrees-of-freedom shear deformable beam element is employed. This element can take into account the
elastic couplings among extension, bending, and torsion arising due to material anisotropy existing in generally
laminated composites as well as the Poisson’s effect. The damage to the beam is introduced by a stiffness loss
coefficient at the elemental level while mass is assumed to be unchanged. Two different objective functions to be
minimized are considered for damage detection. The efficacy of the proposed method with and without the
presence of noise is demonstrated by two numerically simulated composite beams including single and multiple
damages. Although its accuracy somewhat decreases in case of multiple minor damages, the proposed method is
successful to detect moderate and severe single and multiple damages in practice. To overcome this weakness, (i)
the algorithm can be hybridized with other metaheuristics, (ii) objective function can be enhanced by combining
it with other vibration characteristics such as curvature, flexibility, etc., or (iii) two- or multi-stage damage
identification methods can be considered.
Keywords
algorithm; Limited vibration data
1. Introduction
Composite materials have been increasingly used in many industries such as civil, mechanical, aerospace, aviation,
and naval engineering. The increasing interest in composites is due to their advantages such as high strength-to-
weight ratio, corrosion resistance, low thermal expansion, design flexibility, and enhanced fatigue life. However,
the damage mechanisms of composites are very complex and not well understood. They are very much prone to
the occurrence of manufacturing and service-related damages such as matrix cracking, delamination, fiber-matrix
debonding, fiber breakage, etc., that can reach a critical size and thereby affect the safety of the structure causing
human life loss (Ghobadi 2017). Thus, early identification and quantification of any possible damage are necessary
to avoid safety issues and to economize the repair costs.
The conventional damage assessment methods such as visual inspection, infrared thermography, acoustics, and
ultrasonic waves are limited to local application and demand the damaged area to be accessible for inspection.
This limitation can be overcome using vibration-based global methods. They are suitable for damage identification
and quantification in the entire structure by just measuring its vibration response at any location (Doebling et al.
1998; Sinou 2009; Das et al. 2016). In general, the first few modes of vibration can be accessible in practice, and
these modes are less sensitive to small damages. Hence, in such conditions, the damage assessment requires good
vibration-based damage diagnostics along with a powerful optimization technique (Gomes et al. 2018b, 2019c).
In recent years, the use of optimization algorithms in damage detection based on vibration data has been
particularly emphasized (Montalvão et al. 2006; Amafabia et al. 2017; Gomes et al. 2019c). In parallel with the
advances in mathematical science and computer technology, many optimization techniques have been developed
and they have been successfully applied to many engineering problems. In optimization-based damage detection,
an objective function based on modal parameters is made extreme under various constraints to minimize the
difference between the target (measured) and predicted (numerical) models. In this way, the numerical model, i.e.,
the finite element (FE) model, provides the best representation of the real structure, and the location and severity
of the damage in the structure can, thus, be determined with good accuracy. Another advantage of the method is
that it allows the FE model to be updated to represent the current state of the structure. Thus, this last state of the
structure can be used as a reference in the next evaluation.
Starting with the Genetic Algorithm (GA) in the early 1970s, many optimization techniques inspired by nature
have been developed such as Particle Swarm Optimization (PSO), Differential Evolution Algorithm (DE),
Artificial Neural Networks (ANN), and others. In the literature, damage detection with such optimization
techniques using vibration data has been successfully applied to steel and reinforced concrete structural systems
(Hao and Xia 2002; Y Cha et al. 2015; Ding et al. 2016; Wei et al. 2018; Cancelli et al. 2020). However, the
interest in the subject of damage detection in composite laminates by optimization is relatively new and an
increasing trend in the research work has been observed in recent years. As mentioned before, the existence of
extremely complex damage mechanisms in composites makes the optimization-based damage detection methods
more advantageous than the classical ones.
Genetic algorithm (GA) is one of the oldest metaheuristic optimization methods for damage detection in laminated
composites. It is an iterative process in which a population undergoes crossovers and mutations over several
generations. Naturally, this makes GA impractical to use in problems that require many function evaluations. To
overcome this deficiency, it has been hybridized with the algorithms such as ANN (Zhang et al. 2013, 2018; Gomes
et al. 2019b), PSO (Vosoughi and Gerist 2014), and Simulated Annealing (SA) (He and Hwang 2007). Unlike
these studies, Gomes et al. (Gomes et al. 2018a) first determined the probably damaged elements with a two-step
method and then performed the damage severity assessment by GA optimization.
Particle swarm optimization (PSO) is another classical metaheuristic preferred for damage detection in laminated
composites. It has been observed that the method gives fast and effective results in determining the location and
size of delamination damages in composites (Hadi 2016). Compared to the GA, PSO is simpler and more effective
as it requires less function evaluation. Its main disadvantage is, however, that it can converge prematurely before
reaching the best result (global optimum). To overcome this shortcoming, various improvements have been made
to the method and satisfactory results have been obtained for damage detection in laminated composites (Rao et
al. 2015; Masoumi et al. 2018; Jebieshia et al. 2020).
Differential evolution algorithm (DE) is a GA-based metaheuristic that can give effective results, especially in
problems involving continuous variables. Vo-Duy et al. (2016) and Dinh-Cong et al. (2017b, a) developed a two-
stage method for damage detection in laminated composite beams and plates. Therein, the possible damaged
elements were first determined based on the changes in strain energy or modal flexibility, and then the stiffness
loss in the identified elements was computed by the DE algorithm. The most important advantage of such two-
stage damage assessment methods is to significantly reduce the computational cost.
Newer metaheuristics have also been used in the literature for damage detection in laminated composites. Khatir
et al. (2018) used Cuckoo-Search (CS) algorithm for detecting bilateral edge crack (notch) in laminated composite
beams. Gomes et al. (2019a) used the Sunflower optimization (SFO) for detecting a hole in a composite plate.
They compared the results with those of GA and showed that these methods are at least as effective as GA, but
faster in locating and quantifying the damages. Barman et al. (2021) proposed a two-stage method for structural
health monitoring of trusses, space frames, and plates utilizing the Bayesian data fusion approach and teaching-
learning-based optimization (TLBO). In the first stage, suspected damaged elements were determined by
employing Bayesian data fusion on the four damage indices based on modal characteristics. Then TLBO was used
in the second stage considering only the suspected damaged elements instead of all elements in the search space,
which directly reduces the cost of computational effort.
Artificial neural network (ANN) is one of the smart computational techniques used for damage detection in
laminated composites. It has been used to identify delaminations based on the changes in natural frequencies
(Zhang et al. 2013, 2016, 2018; Khan et al. 2019). However, the need for a large amount of data for the learning
process in ANN prolongs the solution time. In addition, the success of the solution largely depends on the accuracy
and diversity of the data used for the learning process. Researchers stated that measurement errors and
environmental noise affect the results negatively in the detection of delamination by ANN.
The above-mentioned metaheuristics have some drawbacks in damage detection applications such as the high
computational cost for large models, premature convergence before reaching a global solution, less accuracy for
multiple damages (in case of the existence of minor damages), and so on. The harmony search algorithm (HSA),
a music-inspired metaheuristic based on the principle of obtaining the best harmonic melody with the notes played
by the musicians in an orchestra (Geem et al. 2001), can successfully handle some of these drawbacks. Compared
to conventional mathematical optimization algorithms, HSA has some advantages such that (i) HSA does not use
binary encoding and decoding but can produce multiple solution vectors, which means it is faster during each
iteration, (ii) HSA imposes fewer mathematical requirements, and does not require setting any initial value for
decision variables. (iii) Since HSA uses stochastic random searches, no derivative information is necessary, (iv)
HSA is less sensitive to the chosen parameters, thus, there is no need for fine-tuning these parameters to get quality
solutions, and (v) the implementation of HSA is easier. In many cases, HSA requires less computational effort for
obtaining the best solution. Thus, the use of HSA and its variants for damage detection in engineering structures
seems to be promising. However, the number of studies using HSA in damage detection is rare. Miguel et al.
(2012) developed a damage detection methodology combining a time-domain modal identification technique (SSI)
with the evolutionary HSA. Jin et al. (2015) presented a new method to investigate the optimal sensor placement
problem on gantry crane structures. They used a combination of an improved HSA and the modal assurance
criterion (MAC).
Since the metaheuristic-based damage detection technique is model-based, it requires complete modal data from
both numerical model and actual structure. Practically, it is not always possible due to the use of a limited number
of sensors on the structure. Therefore, a model reduction technique is necessary to compare the numerical model
and the actual (target) one in optimization. For the present study, Guyan’s method is used.
To the authors’ knowledge, there is no study encountered using HSA for damage detection in laminated
composites. Its advantages over others arouse curiosity about the performance of HSA for damage detection in
laminated composites. Based on this motivation, a numerical procedure to detect and quantify damages in
laminated composite beams with arbitrary lay-up using limited vibration data and HSA is presented here. Since
vibration characteristics of a structure significantly change due to the existence of damages, they can be considered
as the damage diagnostic parameters with the help of an optimization method. For the present study, two different
objective functions to be minimized, one is based on the frequency change while the other is based on the combined
effect of frequency and mode shape changes, are considered. For finite element (FE) analyses, a five-node, 13-
degrees-of-freedom shear deformable beam element is employed. This element can take into account the elastic
couplings among extension, bending, and torsion arising due to material anisotropy as well as the Poisson’s effect
(Kahya et al. 2019). The damage to the beam is introduced by a stiffness loss coefficient at the elemental level
while mass is assumed to remain unchanged. The efficacy of the proposed method with and without the presence
of noise is demonstrated by two numerically simulated composite beams including single and multiple damages.
More precisely, the main contributions of the present study can be summarized as follows: (i) an optimization-
based damage detection method using limited modal data and HSA is developed; (ii) two objective functions,
frequency-based and combined frequency and mode shape-based, are considered to minimize; (iii) damage
detection of laminated composite beams using the proposed method and incomplete modal data is performed; and
(iv) a performance analysis study is carried out to show the method’s efficiency.
The remaining parts of this article are organized as follows: Section 2 briefly introduces the theoretical background
of the shear-deformable finite element model with the definition of damage to the structure, the harmony search
algorithm, and Guyan’s method for model reduction. Section 3 explains the damage detection methodology
proposed. Damage assessment results for assumed composite beams, comparisons with available work, and
performance evaluation of the method are presented and discussed in Section 4. Finally, the concluding remarks
are given in Section 5.
2. Theoretical background
2.1. Shear-deformable finite element model
A schematical representation of a laminated composite beam with a rectangular cross-section is depicted in Fig. 1.
The beam consists of N orthotropic laminae with different fiber orientations and equal thickness. The displacement
field for an orthotropic lamina is expressed based on the first-order shear deformation theory (FSDT) as follows:
0
0
u x z t u x t z x t
v x z t z x t
w x z t w x t

= +
= =
(1)
where u, v, and w denote the displacement components in x-, y-, and z- directions, respectively. u0 and w0 are the
extension and deflection at z = 0, and are the rotations of the normal to the midplane about the y- and x-axis,
respectively. t denotes time. The displacement field in Eqs. (1) considers the effect of shear and rotational inertia
as well as the bending-extension, bending-torsion, and extension-torsion couplings due to material anisotropy.
Based on Eqs. (1), the non-zero strain components can be obtained as follows:
0
where, ,x x = and ,xy x
= denote the normal strain and the bending and torsion curvatures at midplane,
respectively. The notation (•),x shows the derivative with respect to the x variable.
Constitutive relations for the laminated composite beam are given by
0
0
x x
y y
xy xy
x x
y y
xy xy

=


(3b)
where Nx, Ny, and Nxy are the in-plane internal force components, Mx and My are, respectively, the bending moments
about the x and y axes, Mxy is the torsion, Qyz and Qxz are the shear forces perpendicular to the midplane. 0
x ,
0
y ,
and xy are the normal and shear strain components, x, y, and xy are the bending and twisting curvatures,
respectively. In Eqs. (3), the laminate stiffness coefficients A, B, and D are given by
Fig. 1 Schematical representation of an N-layer composite beam with rectangular cross-section
1
1
2
1
1
k
A K Q dz i j
+
+
=
=
= = =
= = =

(4)
where K is the shear correction factor, n is the number of layers in the laminate, and k
ij Q are the transformed
reduced stiffness constants for the kth layer, which can be found in (Reddy 1997).
Due to the Poisson’s effect in the beam, the lateral strains 0
y and
yz are non-zero whereas the force components
Ny and Nxy, the bending moment My, and the shear force Qyz are zero. Therefore, the constitutive relations in Eqs.
(3) can be reduced in the following:
0
55xz xz Q KA = (5b)
where the barred material stiffness coefficients seen above are defined as follows:
1 T
− = −
A = − (6b)
The coefficients A , B and D can be, respectively, replaced by the laminate stiffness coefficients A, B, and D to
neglect the Poisson’s effect.
The equations of motion are obtained by using the Lagrange’s equation given by
0 i i
d L L
dt q q
(7)
where L = T – U is Lagrangian, U and T are, respectively, the strain and kinetic energies defined by
0
0
L h
T u v w bdzdx

−
= + + +
= + +

(8)
where ρ is density and overdot denotes time derivative. In Eq. (7), the term q denotes the unknown field variables.
Substituting Eqs. (8) into Eq. (7) with considering Eqs. (1), (2), and (5) yield the equations of motion in the form
of
+MU KU = 0 (9)
where M and K denote the mass and stiffness matrices, respectively, and the vector U includes the unknowns.
If denotes the natural frequencies and X is the corresponding mode shapes vectors, the solution of Eq. (9) can
be assumed as
i t e
=U X (10)
where 1i = − . Using Eq. (10) into Eq. (9) yields the following standard eigenvalue problem:
2( )− =K M X 0 (11)
The non-trivial solution of Eq. (11) is possible if the condition 2det( ) 0−K M = is satisfied. This gives the natural
frequencies (eigenvalues) of the composite beam. Back substitution of the eigenvalues into Eq. (11) results in the
corresponding eigenvectors (mode shapes).
In Fig. 2, the five-node, 13 degrees-of-freedom (DOFs) finite beam element is depicted. The DOFs at the internal
nodes are for well-representing the complex behavior of anisotropic composite laminates. The unknown field
variables are assumed as
x t x t
(12)
where (x) and (x) are the second and third-order Lagrange polynomials, respectively. Such a selection of the
polynomials for 0u and 0w is for avoiding the shear locking.
Substituting the solutions in Eq. (12) into the equations of motion with considering the definitions given by Eqs.
(1) to (8) yields the following for eth element in the system:
e e e e +M u K u = 0 (13)
Fig. 2 Thirteen-DOF beam element
where Me and Ke are, respectively, the elemental mass and stiffness matrices, ue nodal displacement vector, and
e u is the acceleration vector. The explicit definitions of these matrices and vectors are given in Appendix.
2.2. Definition of damage
Damage in the composite beam is introduced by a stiffness loss coefficient e for each damaged element. Here, it
is assumed that the elemental mass matrix does not change due to damage. For the eth damaged element, the
elemental stiffness is defined as
(1 ) (0 1)D U
e e e e = − K K (14)
where the superscripts D and U denote damaged and undamaged states, respectively. If e = 0, there is no damage
in the corresponding element, and if e = 1, it is fully damaged. Thus, the global mass and stiffness matrices in
Eq. (9) can be obtained in the usual way of the FE method in the form U
e = M M and (1 ) U
e e = −K K ,
improvisation process where the musicians improvise the pitch of their instruments to find the best harmony (Geem
et al. 2001; Yang 2009; Manjarres et al. 2013). The pseudo-code of the HSA is given by Algorithm 1. The
procedure of HSA is briefly described in the following:
Step 1. Initializing the problem and HS parameters: The optimization problem is defined as minimizing (or
maximizing) ( )f x such that L U
i i i x x x where ( )f x is the objective function, 1 2{ }
D x x x=x is a
candidate solution vector consisting of D decision variables, and L
i x
i x are the lower and upper bounds for each
decision variable, respectively. The parameters specified in this step are the harmony memory size (HMS),
harmony memory consideration rate (HMCR), pitch-adjusting rate (PAR), pitch…